Multiplicity Results for Nonlocal Elliptic Transmission Problem with Variable Exponent

: In this paper, a transmission problem given by a system of two nonlin- ear equations of p(x)-Kirchhoﬀ type with nonstandard growth conditions are studied. Using the mountain pass theorem combined with the Ekeland’s variational principle, we obtain at least two distinct, non-trivial weak solutions.


Introduction
Problems related to PDEs involving variable exponents became popular a recently due to their applications and research developments in the modelling of electrorheological fluids, elasticity problems, image processing, mathematical description of the processes filtration of an ideal barotropic gas through a porous medium, etc.; see for example [3,4,9,14,22] and references therein. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field, see [22,23]. The functional analytical tools needed for the analysis have been extensively developed, see [10,11,15] and references therein.
Transmission problems arise in several applications in physics and biology. For instance, one of the important problems of the electrodynamics of solid media is the electromagnetic process research in ferromagnetic media with different dielectric constants. These problems appear as well as in solid mechanics if a body consists of composite materials. We refer the reader to [21] for nonlinear elliptic transmission problems, to [16] for a nonlinear nonlocal elliptic transmission problem. Furthermore, uniqueness and regularity of the solutions to the thermoelastic transmission problem were investigated in [17]. As shown the figure below, let Ω be smooth bounded domain of R n , n ≥ 2, and let Ω 1 ⊂ Ω be a subdomain with smooth boundary Σ satisfying Ω 1 ⊂ Ω. Writing Γ = ∂Ω and Ω 2 = Ω \ Ω 1 we have Ω = Ω 1 ∪ Ω 2 and ∂Ω 2 = Σ ∪ Γ .
In this paper, we are interested in the multiplicity of solutions for nonlocal elliptic systems of gradient type with nonstandard growth conditions. More precisely, we consider the following system with the transmission conditions (1.1) where p ∈ C(Ω), η is outward normal to Ω 2 and is inward to Ω 2 . M and N : R + → R are continuous functions. We confine ourselves to the case where M = N for simplicity. Notice that the results of this paper remain valid for M = N by adding some slight changes in the hypothesis (M 1 ) and (M 2 ) below.
We note that problem (P) with the transmission condition is a generalization of the stationary problem of two wave equations of Kirchhoff type, which models the transverse vibrations of the membrane composed by two different materials in Ω 1 and Ω 2 . Controllability and stabilization of transmission problems for the wave equations can be found in [19,20]. We refer the reader to [1] for the stationary problems of Kirchhoff type, to [6] for elliptic equation p-Kirchhoff type, and to [7,8] for p(x)-Kirchhoff type equation.
In the sequel, we will assume that M : R + → R + is a continuous function satisfying the following assumptions: (M 1 ) There exist real numbers m 1 , m 2 > 0 and θ > 0 such that Although a natural extension of the theory, the problem addressed here is a natural continuation of recent papers. In [5], the authors treat and show the existence of nontrivial solution in the case f (x, u) = λ 1 |u| q(x)−2 u and g(x, v) = λ 1 |v| q(x)−2 v when, λ 1 , λ 2 > 0, q ∈ C(Ω) and 1 < q(x) < p * (x) for all x ∈ Ω, where p * (x) = np(x) n−p(x) if p(x) < n or p * (x) = ∞ otherwise. Motivated by above and the ideas introduced in [13], in this work, we will study the existence of multiple solutions for problem (P) in a more general case when the nonlinear terms f and g are defined by and the following conditions hold: (A) a : Ω 1 → R, satisfies a ∈ L α0(x) (Ω 1 ) and α 0 ∈ C + (Ω 1 ) such that for all x ∈ Ω 1 .
Here, any solution of problem (P) with the transmission conditions will belong to the framework of the generalized Sobolev space, which will be briefly described in the following section, Now, we are ready to state our main result. are satisfied. Then, there exists λ * > 0 such that for any λ 1 + λ 2 ∈ (0, λ * ), problem (P) has at least two distinct non trivial weak solutions.
The paper consists of three sections. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we give the proof of our main result.

Preliminary results and notations
In order to guarantee the integrity of the paper, we recall some definitions and basic properties of variable exponent Lebesgue-Sobolev spaces. For details, we refer to [10,11,15] for the fundamental properties of these spaces.
Let σ : Ω → R be a measurable real function such that σ(x) > 0 for a.e. x ∈ Ω. For p ∈ C + (Ω), define the weighted variable exponent Lebesgue space Equipped with the so-called Luxemburg norm σ(x) (Ω) becomes a separable, reflexive and Banach space. In particular, when An important role in manipulating the generalized Lebesgue spaces is played by the mapping ρ p(x),σ(x),Ω : L The following proposition illuminates the close relation between the |u| p(x),σ(x),Ω and the convex modular ρ p(x),σ(x),Ω . Proposition 2.4. (see [18]) Let p and r be measurable functions such that p ∈ L ∞ (Ω) checking 1 ≤ p(x)r(x) ≤ ∞ for a.e. x ∈ Ω. Then, for u ∈ L r(x) (Ω) with u = 0, the following relations hold.
In particular, if p(x) = p is a constant, then u As in the constant exponent case, for any positive integer k, set Endowed with the norm the space W k,p(x) (Ω) becomes a separable, reflexive and Banach space. In W 1,p(x) 0 (Ω), which denote the closure of C ∞ 0 (Ω) in W k,p(x) (Ω), the Poincaré inequality holds ( [10]), that is, there exists a positive constant C such that (Ω).

Proof of the main result
For simplicity, we use the letters c, c i , i = 1, 2, ... to denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. ., . denote the dual pair.
The energy functional associated to problem (P) is defined as J : E → R, In a standard way, it can be proved that J ∈ C 1 (E, R). Moreover, we have, for all (ϕ, ψ) ∈ E, Thus, weak solutions of problem (P) are exactly the critical points of the functional J. Due to the conditions (M 1 ) and (1.2), we can show that J is weakly lower semicontinuous in E.
In order to establish Theorem 1.2, we need the following lemmas which play an important role in our arguments.
First, we show that {(u n , v n )} is bounded in E. Assume by contradiction the contrary. Then, passing eventually to a subsequence, still denoted by (u n , v n ), we may assume that (u n , v n ) → ∞. Thus, we may consider that u n p(x),Ω1 , v n q(x),Ω2 > 1 for any integer n. Using (M 1 ), (M 2 ) and (3.5), we deduce from (3.8) that Taking into account of 2.1, since α + < θp − and p + < θp + < β − , dividing the above inequality by (u n , v n ) and passing to the limit as n → ∞ we obtain a contradiction. It follows that the sequence {(u n , v n )} is bounded in E. Thus, there exists (u, v) ∈ E such that passing to a subsequence, still denoted by {(u n , v n )}, it converges weakly to (u, v) in E. By (1.2) and the conditions (A) and (B), the embedding from E to the weighted spaces L α(x) (Ω 1 , a(x)) and L β(x) (Ω 2 , b(x)) are compact. Then, using the Hölder inequalities, Propositions |∇v n | p(x)−2 ∇v n (∇v n − ∇v)dx = 0.
Eventually, by [12,Theorem 3.11], we get that (u n , v n ) converges strongly to (u 1 , v 1 ) in E, so we conclude that functional J satisfies the Palais-Smale condition.